How do I solve csc^2x-2cscx = 2 - 4sinx for [0,2pi)?

Question

$csc^2x-2cscx = 2 - 4sinx$ for $[0, 2pi)$

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Answer 1 · 2018-05-19T01:12:52

$csc^2x-2cscx = 2 - 4sinx$

$=>(csc^2x-2cscx) - (2 - 4sinx)=0$

$=>csc^2x(1-2/cscx) - 2(1 - 2sinx)=0$

$=>csc^2x(1-2sinx) - 2(1 - 2sinx)=0$

$=>(csc^2x - 2)(1 - 2sinx)=0$

So $csc^2x=2$

$=>sinx=pm1/sqrt2$

When $sinx=1/sqrt2=sin(pi/4)=sin (pi-pi/4)$

$=>x=pi/4 or (3pi)/4$

When $sinx=-1/sqrt2==sin (pi+pi/4)=sin(2pi-pi/4)$

$=>x=(5pi)/4 or (7pi)/4$

Again when

$1-2sinx=0$

$=>sinx=1/2=sin(pi/6)=sin(pi-pi/6)$

$=>x=pi/6or(5pi)/6$